Editing Moore–Penrose inverse (section)

===Basic properties===
Proofs for the properties below can be found at [[b:Topics in Abstract Algebra/Linear algebra]].
* If {{tmath| A }} has real entries, then so does {{tmath| A^+ }}.
* If {{tmath| A }} is [[invertible matrix|invertible]], its pseudoinverse is its inverse. That is, <math>A^+ = A^{-1}</math>.<ref name="SB2002">{{Cite book | last1=Stoer | first1=Josef | last2=Bulirsch | first2=Roland | title=Introduction to Numerical Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0-387-95452-3 | year=2002}}.</ref>{{rp|243}}
* The pseudoinverse of the pseudoinverse is the original matrix: <math>\left(A^+\right)^+ = A</math>.<ref name="SB2002" />{{rp|245}}
* Pseudoinversion commutes with transposition, complex conjugation, and taking the conjugate transpose:<ref name="SB2002" />{{rp|245}} <!-- reference only mentions the last bit --> <math display="block">\left(A^\mathsf{T}\right)^+ = \left(A^+\right)^\mathsf{T}, \quad \left(\overline{A}\right)^+ = \overline{A^+}, \quad \left(A^*\right)^+ = \left(A^+\right)^* .</math>
* The pseudoinverse of a scalar multiple of {{tmath| A }} is the reciprocal multiple of {{tmath| A^+ }}:<math display="block">\left(\alpha A\right)^+ = \alpha^{-1} A^+</math> for {{tmath| \alpha \neq 0 }}; otherwise, <math>\left(0 A\right)^+ = 0 A^+ = 0 A^\mathsf{T}</math>, or <math>0^+=0^\mathsf{T}</math>.
* The kernel and image of the pseudoinverse coincide with those of the conjugate transpose: <math>\ker\left(A^+\right) = \ker\left(A^*\right)</math> and <math>\operatorname{ran}\left(A^+\right) = \operatorname{ran}\left(A^*\right)</math>.

====Identities====
The following identity formula can be used to cancel or expand certain subexpressions involving pseudoinverses:
<math display="block">
A = {}A{}A^*{}A^{+*}{} = {}A^{+*}{}A^*{}A.
</math>
Equivalently, substituting <math>A^+</math> for <math>A</math> gives
<math display="block">
A^+ ={}A^+{}A^{+*}{}A^*{} = {}A^*{}A^{+*}{}A^+,
</math>
while substituting <math>A^*</math> for <math>A</math> gives
<math display="block">
A^* ={}A^*{}A{}A^+{}={}A^+{}A{}A^*.
</math>